What does the "unused" output on the Vector Math node do?
The answer can be found in the source which isn't hard to follow.

For add and subtract the Value output is the average of the absolute vector values. That is:
$$
\text{value} = \frac{\left\vert v_x\right\vert + \left\vert v_y\right\vert + \left\vert v_z\right\vert}{3}
$$

For average, cross product and normalize it is the length of the vector output.

For dot product the vector output is set to zero.